# Why is the $SU(4)$ group not suitable for describing color symmetry?

How to demonstrate that the $$SU(4)$$ group cannot be a group of symmetry of a color charge?

Simplest way? The $$\Delta^{++}\sim uuu$$ has to be a color singlet. It has spin 3/2, so it is flavor and spin symmetric. But fermion quarks need to be in a fully antisymmetrized state. Can you make an SU(4) singlet out of three antisymmetrized copies of an SU(4) representation, the way you can for SU(3)? (No.)
(By now, you simply experimentally check R in $$e^+e^-$$, which may discriminate between 3 and 4.)